Classification Decisions in Pattern Recognition

The basic element in the solution of pattern-recognition problems is the requirement for the ability to recognize membership in classes. This report considers the automatic establishment of decision criteria for measuring membership in classes that are known only from a finite set of samples. Each sample is represented by a point in a suitably chosen, finite-dimensional vector space in which a class corresponds to a domain that contains its samples. Boundaries of the domain in the vector space can be expressed analytically with the aid of transformations that cluster samples of a class and separate classes from one another. From these geometrical notions a generalized discriminant analysis is developed which, as the sample size goes to infinity, leads to decision-making that is consistent with the results of statistical decision theory. A number of special cases of varying complexity are worked out. These differ from one another partly in the manner in which the operation of clustering samples of a class and the separation of classes is formulated as a mathematical problem, and partly in the complexity of transformations of the vector space which is permitted during the solution of the problem. The assumptions and constraints of the theory are stated, practical considerations and some thoughts on machine learning are discussed, and an illustrative example is given for the automatically learned recognition of spoken words.